Write the conversation response. See the 2 person dialog:
 -Are all hyperkähler manifolds orientable?.
 -Note that Sp("n") ⊂ SU(2"n") ⊂ U(2"n") ⊂ SO(4"n"), so every hyperkähler manifold is a Calabi–Yau manifold, every Calabi– Yau manifold is a Kähler manifold, and every Kähler manifold is orientable..
 -What does it mean that they are orientable?.
 -The strange list above was explained by Simons's proof of Berger's theorem..
 -Are there any other interesting aspects about this article?.
 -A simple and geometric proof of Berger's theorem was given by Carlos E. Olmos in 2005. One first shows that if a Riemannian manifold is "not" a locally symmetric space and the reduced holonomy acts irreducibly on the tangent space, then it acts transitively on the unit sphere..
 -What can you tell me about that proof?.
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The Lie groups acting transitively on spheres are known: they consist of the list above, together with 2 extra cases: the group Spin(9) acting on R, and the group "T" · Sp("m") acting on R. Finally one checks that the first of these two extra cases only occurs as a holonomy group for locally symmetric spaces (that are locally isomorphic to the Cayley projective plane), and the second does not occur at all as a holonomy group.